Guest post from Stephen Wilde:
Greenhouse Gases and the Ideal Gas Law
Stephen Wilde – Jan 2013
This is the usual form of the Ideal Gas Law:
where P is the pressure of the gas, V is the volume of the gas, n is the amount of substance of gas (also known as number of moles), T is the temperature of the gas and R is the ideal, or universal, gas constant, equal to the product of Boltzmann’s constant and Avogadro’s constant.
The Ideal Gas Law as set out above is a representation of certain physical relationships and is therefore not about absolute values.
It is widely known how the various terms within that equation respond to changes in any one or more of them,
P and V are inversely proportional to each other so a rise in Pressure results in reduced Volume and vice versa.
Increasing either P or V without reducing the other requires an increase in
n – total atmospheric mass and/or
R – the gas constant which is related to the strength of the gravitational field and/or
T – Temperature.
The product of n, R and T then rises to match the increased product of P and V.
The problem with AGW theory in relation to the Ideal Gas Law.
AGW theory proposes that an increase in GHGs causes an increase in T which then causes an equal increase in V so as to keep the two sides of the equation balanced.
So far so plausible.
However, an increase in V results in a reduction of density (n) throughout an atmosphere which must REDUCE the product of nRT.
Normally a reduction in n would be accompanied by a reduction in P as well because less mass in an atmosphere results in reduced pressure at the surface if the strength of the gravitational field stays the same but there is no reduction in P at the surface from mere expansion even though the density of the entire atmosphere reduces when V increases.
Therefore we cannot look to a reduction in P to correct the imbalance caused by the reduction in density (n).
According to the Ideal Gas Law it is not possible for PV to fail to equal nRT yet that is just what happens if one holds P steady whilst increasing T and V equally but reducing n.
There would only be balance with PV continuing to equal nRT :
if more mass were added to the atmosphere so as to avoid reducing the average density of the atmosphere when expansion occurred.
or, if the strength of the gravitational field increases to pull V back down to restore n to the previous value
Since no extra mass or gravity is being added AGW theory cannot be right because of the residual imbalance.
If a higher T from more GHGs leads to a higher V then the reduction of density (n) results in lower V which must be accompanied by lower T so there is a logical impasse.
That is what I think some sceptics mean when they say that AGW theory is impossible or contrary to the Laws of physics.
In order to resolve the problem we need to look at the same scenario differently.
What is the effect of adding more GHGs ?
We have proof that GHGs expand the region of an atmosphere in which they are situated.
Since they absorb more energy than non GHGs they spread energy more evenly across the whole area that they occupy. The effect is to reduce the rate at which temperature would otherwise decline with height (the lapse rate).
In the Troposphere the dry adiabatic lapse rate without water vapour would be about 10C per km. The presence of water vapour reduces the actual lapse rate to about 6.5C. As a consequence of reducing the lapse rate the distance required for the air to cool between surface and tropopause needs to increase and so the expanded troposphere pushes the tropopause upwards thereby expanding the troposphere beyond the height that it would have achieved without GHGs.
We see the same process in the stratosphere where ozone (a GHG) warmed by the sun actually reverses the lapse rate so that temperature increases with height up to the stratopause. The expansion of the stratosphere can push both up and down because there is no solid surface beneath it and that results in some interesting features of our climate system that are beyond the scope of this article.
So the effect of GHGs is to increase atmospheric height AND reduce the slope of the lapse rate.
As I will now explain, that is important because the combination of expansion and increased height enables the atmosphere to accommodate more GHGs without altering system equilibrium temperature.
How to approach the problem.
Note first that a constant flow of new energy from outside the atmosphere is required to maintain it in gaseous form.
If that supply of energy from external sources were to be cut off the atmosphere would simply collapse and freeze to the surface in solid form.
Those energy sources can be anything such as from a nearby sun, from geothermal energy below the surface, from nearby planetary gas giants large enough to radiate, even the temperature of space being above absolute zero makes some contribution.
Above all it must be constant because energy is also being radiated to space at an equal rate when the atmosphere is at equilibrium.
Note second that a large amount of work is constantly being done in order to keep the gases lifted off the surface against the constant force of gravity. If the work rate drops the atmosphere will contract and if the work rate increases the atmosphere will expand.
The persistence of a gaseous atmosphere despite the efforts of gravity to pull it down to the surface is due to work being done constantly.
Now, recall the problem we had with AGW theory in that we had nothing to counter the reduction in density (n) caused by more GHGs and we needed something to counter it in order to comply with the Ideal Gas Law.
The rise in V on one side of the equation offset the rise in T on the other side of the equation but we couldn’t balance the numbers because n had reduced on one side of the equation but P had been held steady on the other side.
We obviously need another variable but all we have left is the Gas Constant (R)
Let’s look at R more closely and see what can be done.
Dimensions of R
From the general equation PV = nRT we get
R = PV/nT or (pressure × volume) / (amount × temperature).
As P is defined as force per unit area, so we can also write the gas equation as
R = [(force/area) × volume] / (amount × temperature).
Again, area is simply (length)2 and volume is equal to (length)3. Therefore,
R = [force / (length)2] (length)3 / (amount × temperature).
Since force × length = work,
R = (work) / (amount × temperature).
The physical significance of R is work per degree per mole. It may be expressed in any set of units representing work or energy.
It turns out that R, the Gas Constant is not really constant at all except in clearly defined circumstances unique to each planet. R is only a constant for a fixed gravitational field, a fixed amount of atmospheric mass and a fixed height of atmosphere. Change any of those features and the amount of work required will also change and the value of R will rise or fall.
The amount of work per degree per mole will be related to the strength of a gravitational field. A stronger such field will require more work per degree per mole and the value of R will increase.
It will also be related to the amount of mass that is available to be raised off the surface. The more atmospheric mass the higher the value of R will need to be in order to lift it all off the surface.
Crucially, it will also be related to the height of the atmosphere because a higher atmosphere requires more work to raise molecules to the greater height against the continuing force of gravity so for a higher atmosphere the value of R will increase.
If the atmosphere expands thereby rising in height the value of R will increase because more work needs to be done in order to raise the molecules higher against the force of gravity.
One can then increase the value of R in the equation nRT which will offset the reduction in n to bring both sides of the equation back into balance.
But there is another step to take.
The final step.
It turns out that it isn’t necessary for GHGs to raise T.
In the Ideal Gas Law Equation T is often taken as simply temperature but it is actually more than that.
T is the amount of energy available from all sources to maintain the constant flow that keeps the atmosphere off the surface.
GHGs do not add to that external energy source. Nor do they detract from it.
So it is wrong to include their thermal characteristics within T.
Instead they go straight to expanding the atmosphere thereby raising V
So how do we balance that on the other side of the equation without raising T?
We have determined that a higher atmosphere requires a higher value of R because more work needs to be done in order to maintain the new higher atmosphere.
So all we have to do is raise R and the product of nRT then balances again with PV at the increased V.
The increase in R is enough to offset both the increase in V on the other side of the equation and the reduction of n on the same side of the equation leading to overall balance without raising T.
What happens to the ‘missing energy’
Since there has been no increase in T the total amount of energy flowing through the atmosphere at equilibrium remains the same as before but the GHGs have absorbed more energy so where is it?
The atmosphere has expanded so the total amount of energy held within the system has obviously increased.
The answer is contained in the fact that the energy held within the system isn’t just kinetic energy which registers as heat. It is also comprised of potential energy which does not register as heat.
The higher the atmosphere is allowed to rise within the gravitational field the more of its energy content takes the form of potential energy.
So the extra energy absorbed by GHGs has all gone to increasing atmospheric height which has converted that initial kinetic energy to potential energy where it will remain until the atmosphere contracts again.
That scenario avoids the problem of imbalance inherent in AGW theory, keeps PV = nRT in balance and explains why any extra energy absorbed by GHGs is no longer available to affect equilibrium temperature.
The higher atmosphere does result in air circulation changes that potentially have a climate impact but that is another story that I have dealt with elsewhere.